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Unformatted text preview: 12.5 Homomorphisms – A Generalization of Colourings 663 Lemma 12.5.15  Let H be a core. If the ˜ H-colouring problem is NP- complete, then so is the H-colouring problem. ut h v j ( c ) ( b ) ( a ) Figure 12.5 Illustrating the sub-indicator construction; (a) a digraph H with a special vertex h ; (b) the sub-indicator J with special vertices j,v ; (c) the result ˜ H of applying the sub-indicator construction with respect to ( J,j,v ) to ( H,h ). To illustrate how to use the indicator and the sub-indicator construction, let us show that, if H is the digraph in Figure 12.4(a), then the H-colouring problem is NP-complete. First apply the indicator construction with respect to the indicator shown in Figure 12.4(b) to H . This gives us the digraph H * in Figure 12.4(c). By Lemma 12.5.14, H-colouring is NP-complete if and only if H *-colouring is NP-complete. Now let J be the sub-indicator consisting of the complete biorientation of a 3-cycle with one vertex labelled j and an isolated vertex v 1 . Let H be the result of applying the sub-indicator construction with respect to ( J,j,v 1 ) to H * . Since v 1 is isolated, a vertex from H * will be in H precisely when it is itself on a complete biorientation of a 3- cycle in H * . Hence H is the complete biorientation of a 3-cycle. By Theorem 12.5.10 H-colouring is NP-complete and now we conclude by Lemma 12.5.15 that H *-colouring and hence also H-colouring is NP-complete. Although the sub-indicator and the indicator constructions are very useful tools for proving the NP-completeness of many H-colouring problems, there are digraphs H for which another approach such as a direct reduction from a different type of NP-complete problem is needed. Such reductions are often from some variant of the satisfiability problem (see Section 1.10). The reader is asked to give such a reduction in Exercise 12.29. For examples of other papers dealing with homomorphisms in digraphs see  by Brewster and MacGillivray,  by Hell, Zhou and Zhu,  by Neˇ setˇ ril and Zhu,  by Sophena and [761, 762] by Zhou. 664 12. Additional Topics 12.6 Other Measures of Independence in Digraphs The definition of independence of vertex subsets in digraphs used in this book is by no means the only plausible definition of independence in digraphs. One may weaken the definition of independence in directed graphs in at least two other ways, both of which still generalize independence in undirected graphs. (1) By considering induced subdigraphs which are acyclic. This gives rise to the acyclic independence number , α acyc ( D ), which denotes the size of a maximum set of vertices X such that D h X i is acyclic. (2) By considering induced subdigraphs which contain no 2-cycles. This gives rise to the oriented independence number , α or ( D ), which denotes the size of a maximum set of vertices Y such that D h Y i is an oriented graph....
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This document was uploaded on 08/10/2011.
- Spring '11